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arXiv · 1911.13006

Full proof of Kwapie\'n's theorem on representing bounded mean zero functions on $[0,1]$

Aleksei F. Ber·Matthijs J. Borst·Fedor A. Sukochev
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Abstract

In [7], Kwapie\'{n} announced that every mean zero function $f\in L_\infty[0,1]$ can be written as a coboundary $f = g\circ T -g$ for some $g\in L_\infty[0,1]$ and some measure preserving transformation $T$ of $[0,1]$. Whereas the original proof in [7] holds for continuous functions, there is a serious gap in the proof for functions with discontinuities. In this article we fill in this gap and establish Kwapie\'{n}'s result in full generality. Our method also allows to improve the original result by showing that for any given $\epsilon>0$ the function $g$ can be chosen to satisfy a bound $\|g\|_\infty\leq (1+\epsilon)\|f\|_\infty$.

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Aleksei F. Ber, Matthijs J. Borst, Fedor A. Sukochev. 2019-11-29. Full proof of Kwapie\'n's theorem on representing bounded mean zero functions on $[0,1]$. https://arxiv.org/abs/1911.13006

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